环上矩阵的Drazin逆(英文)

The Drazin Inverse of Matrices over Rings ①

设Ｒ是一交换环，Ａ∈Ｍｎ（Ｒ），定义Ａ的指数和Ｄｒａｚｉｎ逆ＡＤ．假设Ａｋ＝ＰＢＱ，ｋ＝ｉｎｄ（Ａ），其中Ｐ，Ｂ，Ｑ∈Ｍｎ（Ｒ），存在Ｐ′，Ｑ′∈Ｍｎ（Ｒ），使得Ｐ′ＰＢ＝Ｂ＝ＢＱＱ′，若Ｂ有群逆，则Ａ有Ｄｒａｚｉｎ逆当且仅当ＢＢ＃ＱＡＰＢ＋Ｉ－ＢＢ＃可逆，此时有ＡＤ＝ＰＢ（ＢＢ＃ＱＡＰＢ＋Ｉ－ＢＢ＃）－１Ｑ．

Let R be a commutative ring,and let A∈M n×m (R).We defined the index and the Drazin inverse of A,SupposeA k=PBQ,for k =ind(A),where P,B,Q∈M n×m (R),and there exist P′,Q′∈M n× m (R) such that P′PB=B=BQQ′,If B has group inverse B #,then A has a Drazin inverse iff BB #QAPB+I-BB # is invertible.In this case,A D=PB(BB #QAPB+I-BB #) -1 Q.